Fractal Interpolation Function for Countable Data

Abstract

The problem of extending the theory of an iterated function system carried over to an infinite and countable iterated function system has been widely studied in the last decades. Such an extension is applied in the theory of sampling and reconstruction where a countable iterated function system contributes to a better approximation. It is notable that interpolation and approximation regularly show up as two sides of the same coin; results about one as often as possible infer results about the other. To substantiate this in the context of the fractal interpolation function, Barnsley perceived that a given continuous function f on a closed interval, say I, in $\mathbb {R}$ to $\mathbb {R}$ can be expressed as a class of fractal functions. This is accomplished by choosing a finite number of points $\{(x_n, y_n): n\in \mathbb {N}_N\}$ on $I\times \mathbb {R}$ and reconstructing f through the fractal interpolation function with an appropriately defined iterated function system. In the conventional setting of fractal interpolation and in practically the entirety of its expansions referenced previously, a fractal interpolation function is developed for a finite data set. As mentioned earlier, there are numerous natural phenomena where an infinite data set might be necessary, for example, in the theory of signal reconstruction and the sampling scheme. This idea inspired to investigate the existence of the fractal interpolation function for an infinite data set instead of a finite number of data points. Recently, the standard construction of the fractal interpolation function reached out from the finite data set to the case of a countable number of data points with the notion of the countable iterated function system [22, 49]. This development of the fractal interpolation function for a prescribed countable set of data frames the reason for the discussion of fractional calculus of the fractal interpolation function for countable data. Hence, this chapter introduces and describes the sequence of data and the corresponding interpolation function which is a generalization of the Secelean framework. The existence of the continuous function f which interpolates the sequence of prescribed data $\left\{ (x_n, y_n): n\in \mathbb {N}\right\}$ is discussed, where $(x_n)_{n=1}^{\infty }$ is a monotonic real sequence and $(y_n)_{n=1}^{\infty }$ is a bounded sequence of real numbers. In addition to that, the prominent influence of free parameters in the shape of a fractal interpolation function is illustrated with suitable examples. Besides, the existence of a countable iterated function system is investigated when the fractal interpolation function for a sequence of data is given. Further, the existence of the Riemann–Liouville fractional integral and derivative of the fractal interpolation function of countable data set (or sequence of points) is established.